线性子空间上具有极大投射维数的多重结构
Multiple Structures with Arbitrarily Large Projective Dimension on Linear Subspaces
摘要：
设$K$是代数闭域。在小余维线性子空间上定义的$¶^n_K$中，在附加假设（如Cohen-Macaulay假设）下刻画多个结构引起了人们极大的兴趣。我们表明，没有这样的有限表征的多个结构是可能的，如果只假设塞尔的$(S_1)$属性举行。具体地说，我们证明了：对任意正整数$h, e \ge 2$，有$(h,e) \neq (2,2)$和$p \ge 5$，在$K$上的多项式环中存在齐次理想$I$，使得（1）$I$的高度是$h$，（2）$R/I$的Hilbert-Samuel重数是$e$，（3）$R/I$的投射维数至少为$p$;（4）理想的$I$是线性素数$(x_1,..., x_h)$的素。这一结果是在鲜明的对比Manolache的表征科恩-麦考利多重结构的余维2和多重性最多4，也Engheta的表征纯理想的高度2和多重性2。
英文摘要：
Let $K$ be an algebraically closed field. There has been much interest in characterizing multiple structures in $¶^n_K$ defined on a linear subspace of small codimension under additional assumptions (e.g. Cohen-Macaulay). We show that no such finite characterization of multiple structures is possible if one only assumes Serre's $(S_1)$ property holds. Specifically, we prove that for any positive integers $h, e \ge 2$ with $(h,e) \neq (2,2)$ and $p \ge 5$ there is a homogeneous ideal $I$ in a polynomial ring over $K$ such that (1) the height of $I$ is $h$, (2) the Hilbert-Samuel multiplicity of $R/I$ is $e$, (3) the projective dimension of $R/I$ is at least $p$ and (4) the ideal $I$ is primary to a linear prime $(x_1,..., x_h)$. This result is in stark contrast to Manolache's characterization of Cohen-Macaulay multiple structures in codimension 2 and multiplicity at most 4 and also to Engheta's characterization of unmixed ideals of height 2 and multiplicity 2.
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